GATE 2022 MA – Question 58

Numerical Analysis · Numerical Initial Value Problems · 2 marks · Numerical answer

For y′=f(x,y), a second-order Runge–Kutta method uses K₁=hf(x_i,y_i),K₂=hf(x_i+αh,y_i+βK₁), y_(i+1)=y_i+(K₁+3K₂)/4. Find αβ to two decimals.

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Correct answer: 0.43 to 0.45

Explanation

Expand K₂=hf+h²(αf_x+βf_yf)+O(h³). The update becomes y+hf+(3/4)h²(αf_x+βf_yf). Match the exact Taylor coefficient (1/2)h²(f_x+f_yf): α=β=2/3. Product=4/9=**0.44**.

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